Teaching Probability

We’re rather proud of our new book Teaching Probability, which is the first in the Cambridge Mathematics series. It also seems to be the first book dedicated to the teaching of probability to 11-16 year-olds, which perhaps reflects the low priority that has been previously attached to the topic. Given the vital importance of probability, it is time for a change in attitude.

So why is probability important? First, it is an intrinsic part of mathematics, with attractive methods to solve challenging abstract problems. But perhaps more important is its relevance to the real world, as it provides the formal framework for dealing with chance, randomness, and uncertainty in general. It also provides the basis for formal statistical inference, although we do not cover this aspect in our book.

Given this importance, it’s reasonable to ask why it is generally not a popular or well-understood part of the curriculum. We can think of a number of possible reasons. First, we should acknowledge that it can be difficult. Problems can often be verbally expressed in simple terms, but getting to a solution is generally not a mechanical process. Even now, when confronted with problems designed for schools, we find we have to stop, sit down quietly, carefully work through a possible solution, and then check it by a different route. And then check it again.

We believe another reason for the lack of popularity of probability is its identification with permutations and combinations. This is deeply unfortunate, since combinatorics are concerned with counting the size of sets, and have nothing intrinsically to do with probability. We pay minimal attention to these topics in our book.

But finally we should admit that probability is not particularly intuitive. When British Members of Parliament were asked “If you spin a coin twice, what is the probability of getting two Heads”, only 40% were able to answer correctly [Spoiler: the answer is ¼]. Things aren’t helped by the ambiguity that is common in casual probability statements: for example, does the statement “the chance of a false-positive doping test” refer to the chance of a random test being a false-positive, a random positive test being a false identification, or a random test on an innocent competitor coming out positive? All are plausible interpretations, and all are quite different. This example shows the vital importance of being clear about the ‘reference class’ – essentially the denominator of the fraction that comprises the probability.

Our solution to these undoubted challenges is to learn from the latest psychological research on communicating risks to the general population, in which it has been shown that rather than telling people, for example, that “their chance of a heart attack or stroke in the next ten years is 12%”, doctors are now taught to say “Out of 100 people like you, we would expect 12 to have a heart attack or stroke in the next ten years”. This change to the language of ‘expected frequencies’ not only makes the reference class completely clear (i.e. 100 people like you), but also has been shown to help with difficult probability questions.

Consider the following problem.

Suppose a screening test for doping in sports is claimed to be ‘95% accurate’, meaning that 95% of dopers, and 95% of non-dopers, will be correctly classified. Assume 1 in 50 athletes are truly doping at any time. If an athlete tests positive, what is the probability that they are truly doping?

The way to answer such questions is to think of what we would expect to happen for every, say, 1000 tests conducted. Out of these, 1 in 50 (20) will be true dopers, of which 95% (19) will be correctly detected. But of the 980 non-dopers, 5% (49) will incorrectly test positive. That means a total of 68 positive tests, of which 19 are true dopers. So the probability that someone who tests positive is truly doping is 19/68 = 28%. So, among the positive tests, the ‘false-positive’ results greatly outnumber the correct detections by around 3 to 1.

If you find it difficult to make sense of these numbers, the expected frequency tree may help to clarify them:

The crucial idea is that by working with expected frequencies instead of probabilities, we have been able to solve a complex conditional probability (Bayes’ theorem) problem.

In the book we show how an entire probability syllabus can be taught by beginning with experimentation with randomness – we advise using spinners rather than dice and coins, and provide a wide range of templates. By collating results over many experiments, the idea of empirical frequencies is introduced, which for two-stage experiments can be represented as Venn diagrams, trees and 2-way tables. The crucial stage is then to ask, given the empirical data and what can be observed about the process, what we would expect to happen in a new series of experiments. Having introduced expected frequencies, and again represented these in a variety of formats, the rules of probability arise naturally rather than being plucked out of the air.

We provide a series of lessons developed in detail, as well as worked solutions to a huge range of potential examination questions. We conclude with a series of extension exercises and projects showing the extraordinary richness of probability in the real world, including lotteries, insurance, risk, weather forecasting and so on.

As we said at the start, we are rather proud of the book and feel it is a genuinely novel, but intensely practical, contribution to maths pedagogy.

David Spiegelhalter is Winton Professor for the Public Understanding of Risk in the Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge and co-author of Teaching Probability.